By Andrei A. Bytsenko, G. Cognola, E. Elizalde, V. Moretti, S. Zerbini

One of many goals of this ebook is to provide an explanation for in a easy demeanour the probably tough problems with mathematical constitution utilizing a few particular examples as a consultant. In all the circumstances thought of, a understandable actual challenge is approached, to which the corresponding mathematical scheme is utilized, its usefulness being duly validated. The authors try and fill the distance that often exists among the physics of quantum box theories and the mathematical tools most fitted for its formula, that are more and more tough at the mathematical skill of the physicist.

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**Additional resources for Analytic Aspects of Quantum Fields**

**Example text**

With residues which are polynomials in fi. E. A. Weldon (1982a)] to compact manifolds then reads n/9(/3,/i)=-^r / dsCR(s)T(s-l)[as-l\Q+) + as-l\Q-)]p-s. ) as a function of complex p. 5 Regularization of the vacuum energy The vacuum energy density (the first term on the right hand side in Eq. 65)) is the only source of divergences in the thermodynamic potential. Therefore, we shall now briefly discuss how to give it a mathematical mean ing. We consider a general formula for the vacuum energy of a scalar field defined on an ultrastatic spacetime with compact spatial section consisting, in general, in a manifold with boundary.

1 Heat-Kernel Expansion and Coefficients The heat-kernel expansion on compact manifolds The zeta function technique is based on several mathematical properties of the heat kernel associated with the operator A. In the following we summarize relevant statements of the heat kernel on a compact manifold. M. Wald (1979); I. B. B. Davies (1989)]. P. do Carmo (1992)]. 9) (the index x in Ax means that A acts on the variable x) with initial condi tion, lim / d/j,g(y) K(t,x,y\A)i{j(y) *-*-o+ JM uniformly in x, for all tp 6 C°(M).

D The function Oi°jy above depends on ^ while the coefficients aj are independent. 31) belongs t o either C°°((0,+oo) x /„ x /„) or C°°((0,+oo) x M x M) for a = P = Q. 19) may not be asymptotic if x / y and t > 0 as N -> oo. Indeed, due to the presence of ?? 19) where the exponential e~a(x'y^2t appears. 19). A. Fulling (1991); R. A. Bytsenko et al. D.